Evaluate $\cfrac{\left\lceil\cfrac{17}{7}-\left\lceil\cfrac{27}{17}\right\rceil\right\rceil}{\left\lceil\cfrac{27}{7}+\left\lceil\cfrac{7\cdot17}{27}\right\rceil\right\rceil}$
Solution: The first thing to be addressed is the fractions under the inner sets of ceiling functions. The smallest integer greater than $\frac{27}{17}$ is $2$. The smallest integer greater than $\frac{7\cdot17}{27}$, which is equal to $\frac{119}{27}$ is $5$. Therefore, the original problem can be rewritten as: \[\frac{\left\lceil\frac{17}{7}-2\right\rceil}{\left\lceil\frac{27}{7}+5\right\rceil}=\frac{\left\lceil\frac{3}{7}\right\rceil}{\left\lceil\frac{62}{7}\right\rceil}\] The smallest integer greater than $\frac{3}{7}$ is $1$ and the smallest integer greater than $\frac{62}{7}$ is $9$. Hence, the final simplified fraction is $\boxed{\frac{1}{9}}$.